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Cross-moments of a random vector

This lecture defines the notion of cross-moment of a random vector, which is a generalization of the concept of moment of a random variable (see the lecture entitled Moments of a random variable).

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Cross-moment

Let X be a Kx1 random vector. A cross-moment of X is the expected value of the product of integer powers of the entries of X:[eq1]where X_i is the i-th entry of X and [eq2] are non-negative integers.

The following is a formal definition of cross-moment.

Definition Let X be a Kx1 random vector. Let [eq3] and [eq4]. If[eq5]exists and is finite, then it is called a cross-moment of X of order n. If all cross-moments of order n exist and are finite, i.e. if (1) exists and is finite for all K-tuples of non-negative integers [eq6] such that [eq7], then X is said to possess finite cross-moments of order n.

The following example shows how to compute a cross-moment of a discrete random vector.

Example Let X be a $3	imes 1$ discrete random vector and denote its components by X_1, X_2 and $X_{3}$. Let the support of X be [eq8]and its joint probability mass function be[eq9]The following is a cross-moment of X of order $4$:[eq10]which can be computed by using the transformation theorem:[eq11]

Central cross-moment

The central cross-moments of a random vector X are just the cross-moments of the random vector of deviations [eq12].

Definition Let X be a Kx1 random vector. Let [eq3] and [eq4]. If[eq15]exists and is finite, then it is called a central cross-moment of X of order n. If all central cross-moments of order n exist and are finite, that is, if (2) exists and is finite for all K-tuples of non-negative integers [eq16] such that [eq7], then X is said to possess finite central cross-moments of order n.

The following example shows how to compute a central cross-moment of a discrete random vector.

Example Let X be a $3	imes 1$ discrete random vector and denote its components by X_1, X_2 and $X_{3}$. Let the support of X be[eq18]and its joint probability mass function be[eq19]The expected values of the three components of X are[eq20]The following is a central cross-moment of X of order $3$:[eq21]which can be computed by using the transformation theorem:[eq22]

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